Remark 3.10 It is well known that
if \(x\in \mathbb {X}\backslash K\)  and \((\mathbb {X},\Vert \cdot \Vert )\)  has a uniformly
Gâteaux differentiable norm, then function \({\rm dist}\left(\cdot ,K\right)\)  is strictly
differentiable at \(x\) , with \({\rm D}{\rm dist}\left(\cdot ,K\right)(x)\in {\mathbb {S}}^*\) 
(see [1]}), so in this case one has
\(\widehat{B}^*_K(x)=\lbrace {\rm D}{\rm dist}\left(\cdot ,K\right)(x)\rbrace \) . The class
of Banach spaces admitting a uniformly Gâteaux differentiable norm
is known to include all separable spaces (see [2]}).

